In 1931, Kurt Gödel determined the incompleteness of formal axiomatic systems by demonstrating that there are propositions that cannot be proved or disproved within the system in question. In 1936, Alan Turing showed that some functions cannot be computed, and thereby described the limits of computing machines before any such machine was built. In this essay I will turn to these logical discoveries in order to argue that incompleteness and incomputability can be employed as conceptual tools to re-engage with the axiomatic character of computation. This re-engagement with formal axiomatic structures, I will claim, has important consequences for the aesthetic investigation of computation. Computational aesthetics is understood here as an enquiry into the relation between abstraction and experience in computation. In this respect, the concepts of the incomplete and the incomputable will be mobilised philosophically, beyond the technical scope of Gödel’s and Turing’s work, to argue for the autonomy and reality of computational abstraction. Gödel’s and Turing’s discoveries preclude the possibility that axiomatic formulation could be the method through which the metacomputation of the intelligible and the sensible is accomplished. These discoveries in fact prove that the computational axiomatic system is not transcendentally closed to contingency, as metacomputational approaches geared towards the formulisation of reality would have it, but is instead immanently open to its own eventuality. I will thus argue that there is a contingent ontology of computation that is to be found within computation’s formalisms. This contingent ontology is not predicated upon the empirical or the phenomenal, but is inherently formal and computational. For the philosophical investigation of the aesthetics of computation, such an ontology of the contingent computational structure opens up the possibility of thinking computational forms beyond the limits of formulae, together with that of engaging with formalism beyond the idealisation of beautiful and truthful determinisms.